@abi_kcuthere: "Hi *,
   I'd like to know the currently best known complexity to solve the
   subgroup problem in free groups.
the problem is as follows. 
Let Z be a finite alphabet, and consider the free group G on {Z U Z'}
where Z' is the set of inverses of the elements of Z. 
if K is a subset of G, and consider the subgroup generated by {K U K'}
where K' is the set of all the inverses of the words in U. We would
like to test membership in the subgroup of G generated by {K U K'}.
I had read about Nielsen reductions, but all this is going above my
head. I'd like to know the complexity of the best known solution to
this problem. I am trying to know whether it is in NP or not and what
is its complexity class.
Thanks,
abi."
    @abi_kcuthere: "actually, from what i read about free groups and todd coxeter
enumeration, and stuff, 
1. The problem of finding whether some word lies in this sub-group is
   decidable only if the index of the subgroup is finite.
2. And it is undecidable in general to compute the index of the
   subgroup.
So, can this be exploited to generate instances of the word problem
that are undecidable?
or am i wrong in my understanding of facts 1 and 2
thanks for any help
bye
abi"
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